Edexcel A-level Further Maths revision notes

Elastic collisions in two dimensions

Section FM1-5

Year 2

2 specification points

Notes and three levels of exam-style practice for each registered specification point in this section.

Checked against Edexcel 9FM0 section FM1-5

Checked against Edexcel 9FM0 section FM1-5. Review basis: the qualification registry sourced from the Pearson Edexcel Level 3 Advanced GCE in Further Mathematics (9FM0) specification; registry verification recorded 17 July 2026.

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FM1-5.1

Oblique impact of smooth elastic spheres and a smooth sphere with a fixed surface. Loss of kinetic energy due to impact.

Notes

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Explanation

For an oblique impact, resolve each velocity into components parallel and perpendicular to the line of centres, or tangential and normal to a fixed surface. Smoothness makes the impulse normal to the contact, so tangential components remain unchanged.
For two equal-radius spheres, apply momentum and Newton's law of restitution only along the line of centres, then recombine the new normal components with the unchanged transverse components.
At a fixed surface, reverse and scale the normal component by $e$ .
Kinetic-energy loss comes only from the changed normal components.
Examiners expect the resolution directions to be shown, signed normal components to be used, and complete final velocity vectors rather than normal components alone.

An oblique approach resolved along and perpendicular to the line of centres.

Worked example

A sphere approaches a fixed smooth wall with velocity $(8\mathbf i+5\mathbf j)\,\text{m s}^{-1}$ , where $\mathbf i$ is normal towards the wall and $\mathbf j$ is tangential. If $e=\tfrac34$ , find the velocity and kinetic-energy fraction remaining after impact.

1.The tangential component remains $5\mathbf j$ .
2.The normal component reverses and becomes $-(\tfrac34)(8)\mathbf i=-6\mathbf i$ .
3.The kinetic-energy fraction is the ratio of speed squares: $\dfrac{6^2+5^2}{8^2+5^2}=\dfrac{61}{89}$ .

Answer: Velocity $(-6\mathbf i+5\mathbf j)\,\text{m s}^{-1}$ ; kinetic-energy fraction remaining $\dfrac{61}{89}$ .

Common mistakes

Don't apply restitution to the whole speed instead of only the normal component.
Don't change the tangential component even though the contact surfaces are smooth.
Don't apply sphere-sphere momentum conservation separately to transverse components affected by no impulse.

Exam tip

Draw the line of centres and resolve every velocity before writing momentum or restitution equations.

Tier 1 · Easy

ORIGINAL

The unit vector

\mathbf i

points normally towards a fixed smooth wall and

\mathbf j

is parallel to it. A sphere arrives with velocity

(3\mathbf i-4\mathbf j)\,\text{m s}^{-1}

and has coefficient of restitution

1/2

with the wall. Find its velocity after impact.

(2)

(Total for Question 1 is 2 marks)

Tier 2 · Standard

ORIGINAL

Two smooth spheres of equal radius and mass

2\,\text{kg}

collide. At impact,

\mathbf i

is directed along their line of centres. Their velocities are

(6\mathbf i+2\mathbf j)\,\text{m s}^{-1}

and

(\mathbf i-3\mathbf j)\,\text{m s}^{-1}

, and

e=1/2

. Find both velocities after impact and the impulse on the first sphere.

(5)

(Total for Question 1 is 5 marks)

Tier 3 · Hard

ORIGINAL

Two smooth spheres have equal radius and each has mass

2\,\text{kg}

. At collision their line of centres is parallel to

\mathbf i

, and their velocities are

(5\mathbf i+4\mathbf j)\,\text{m s}^{-1}

and

(-\mathbf i+2\mathbf j)\,\text{m s}^{-1}

. Given

e=1/3

, determine both velocities after impact and the loss of kinetic energy.

(6)

(Total for Question 1 is 6 marks)

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FM1-5.2

Successive oblique impacts of a sphere with smooth plane surfaces.

Notes

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Your confidence:

Explanation

At each smooth plane, retain the tangential velocity component and reverse the normal component with its magnitude multiplied by that plane's coefficient of restitution. The complete post-impact vector becomes the pre-impact vector for the next plane, and different surfaces may have different coefficients.
Geometry and direction determine the order of impacts, so the updated vector must actually point towards the stated next surface.
In a right-angled corner, successive impacts can alter perpendicular components independently.
Speed and kinetic energy usually decrease because normal components are scaled by numbers in $[0,1]$ , while a component parallel to a plane is unchanged at that impact.
Examiners expect one labelled vector after each collision and a direction check.

Worked example

A sphere in a right-angled corner has velocity $(12\mathbf i+5\mathbf j)\,\text{m s}^{-1}$ . It strikes the wall normal to $\mathbf i$ with $e=\tfrac12$ , then the wall normal to $\mathbf j$ with $e=\tfrac35$ . Find its final velocity and the fraction of initial kinetic energy remaining.

1.After the first wall, the velocity is $-6\mathbf i+5\mathbf j$ .
2.After the second wall, the velocity is $-6\mathbf i-3\mathbf j$ .
3.The energy fraction is the ratio of speed squares: $\dfrac{36+9}{144+25}=\dfrac{45}{169}$ .

Answer: Final velocity $(-6\mathbf i-3\mathbf j)\,\text{m s}^{-1}$ ; fraction remaining $\dfrac{45}{169}$ .

Common mistakes

Don't start the second impact from the original velocity rather than the first post-impact velocity.
Don't use the first surface's coefficient of restitution for both plane impacts.
Don't reverse both velocity components at a smooth plane instead of only its normal component.

Exam tip

Write a short vector sequence, one line per plane, so that each unchanged tangential component is visible.

Tier 1 · Easy

ORIGINAL

A sphere approaches a vertical smooth wall with velocity

(6\mathbf i+8\mathbf j)\,\text{m s}^{-1}

, where

\mathbf i

is normal to the wall. The coefficient of restitution is

1/2

. Find its velocity and speed immediately after impact.

(3)

(Total for Question 1 is 3 marks)

Tier 2 · Standard

ORIGINAL

A sphere moves inside a right-angled corner with velocity

(8\mathbf i+6\mathbf j)\,\text{m s}^{-1}

. It strikes first the wall normal to

\mathbf i

and then the wall normal to

\mathbf j

. Its coefficient of restitution with each wall is

1/2

. Find its velocity after the second impact and the fraction of its initial kinetic energy that remains.

(5)

(Total for Question 1 is 5 marks)

Tier 3 · Hard

ORIGINAL

A sphere travels inside a right-angled corner with velocity

(10\mathbf i+6\mathbf j)\,\text{m s}^{-1}

. It hits the wall normal to

\mathbf i

with coefficient of restitution

e

, then the wall normal to

\mathbf j

with coefficient

1/2

. After both impacts its direction is

30^\circ

below the negative

\mathbf i

direction. Find

e

and the fraction of its initial kinetic energy lost.

(6)

(Total for Question 1 is 6 marks)

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